Hormander class of pseudo-differential operators on compact Lie ...

¨ HORMANDER CLASS OF PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS AND GLOBAL HYPOELLIPTICITY

arXiv:1004.4396v1 [math.FA] 26 Apr 2010

MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH Abstract. In this paper we give several global characterisations of the H¨ ormander class Ψm (G) of pseudo-differential operators on compact Lie groups. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudodifferential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.

1. Introduction 1.1. For a compact manifold M we denote by Ψm (M) the set of H¨ormander’s pseudodifferential operators on M, i.e., the class of operators which in all local coordinate charts give operators in Ψm (Rn ). Operators in Ψm (Rn ) are characterised by the symbols satisfying |∂ξα ∂xβ a(x, ξ)| ≤ C(1 + |ξ|)m−|α| for all multi-indices α, β and all x, ξ ∈ Rn . An operator in Ψm (M) is called elliptic if all of its localisations are locally elliptic. The principal symbol of a pseudo-differential operator on M can be invariantly defined as function on the cotangent bundle T ∗ M, but it is not possible to control lower order terms in the same way. If one fixes a connection, however, it is possible to make sense of a full symbol, see Widom [18]. However, on a Lie group G it is a very natural idea to use the Fourier analysis and the global Fourier transform on the group to analyse the behaviour of operators which are originally defined by their localisations. This allows one to make a full use of the representation theory and of many results available from the harmonic analysis on groups. In [9] and [7], the first two authors carried out a comprehensive non-commutative analysis of an analogue of the Kohn–Nirenberg quantisation on Rn , in terms of the representation theory of the group. This yields a full matrix-valued b which can be viewed as a non-commutative symbol defined on the product G × G, version of the phase space. The calculus and other properties of this quantisation resemble those well-known in the local theory on Rn . However, due to the noncommutativity of the group in general, the full symbol is matrix-valued, with the b equal to the dimension of dimension of the matrix symbol a(x, ξ) at x ∈ G and [ξ] ∈ G

Date: April 27, 2010. 2010 Mathematics Subject Classification. Primary 35S05; Secondary 22E30. Key words and phrases. Pseudo-differential operators, compact Lie groups, microlocal analysis, elliptic operators, global hypoellipticity, Leibniz formula. The first author was supported by the EPSRC Leadership Fellowship EP/G007233/1. The third author was supported by the EPSRC grant EP/E062873/1. 1

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MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

the representation ξ. This construction resembles that of Taylor [15] but is carried out entirely in terms of the group G without referring to its Lie algebra and the Euclidean pseudo-differential classes there. In [2] (see also [16]), pseudo-differential operators have been characterised by their commutator properties with the vector fields. In [7], we gave a different version of this characterisation relying on the commutator properties in Sobolev spaces, which led in [9] (see also [7]) to characterisations of symbols of operators in Ψm (G). However, the results there still rely on commutator properties of operators and some of them are taken as assumptions. One aim of this paper is to eliminate such assumptions from the characterisation, and to give its b This will different versions relying on different choices of difference operators on G. be given in Theorem 2.2. In Theorem 2.6 we give another simple description of H¨ormander’s classes on the group SU(2) and on the 3-sphere S3 . This is possible due to the explicit analysis carried out in [9] and the explicit knowledge of the representation of SU(2). We note that this approach works globally on the whole sphere, e.g. compared to the analysis of Sherman [11] working only on the hemisphere (see also [12]). Using the classification of representations in some cases (see, e.g., [10]) it is possible to draw conclusions of this type on other groups as well. However, this falls outside the scope of this paper. We give two applications of the characterisation provided by Theorem 2.2. First, we characterise elliptic operators in Ψm (G) by their matrix-valued symbols. Similar to the toroidal case (see [1]) this can be applied further to spectral problems. Second, we give a sufficient condition for the global hypoellipticity of pseudo-differential operators. Since the hypoellipticity depends on the lower order terms of an operator, a knowledge of the full symbol becomes crucial. Here, we note that while classes of hypoelliptic symbols can be invariantly defined on manifolds by localisation (see, e.g., Shubin [13]), the lower order terms of symbols can not. From this point of view conditions of Theorem 5.1 appear natural as they refer to the full symbol defined b globally on G × G. The global hypoellipticity of vector fields and second order differential operators on the torus has been extensively studied in the literature, see e.g. [3], [4], and references therein. In this paper we study the global hypoellipticity problem for pseudo-differential operators on general compact Lie groups, and discuss a number of explicit examples on the group S3 ∼ = SU(2). We say that an operator A is globally hypoelliptic on G if Au = f and f ∈ C ∞ (G) imply u ∈ C ∞ (G). Thus, we give examples of first and second order differential operators which are not locally hypoelliptic and are not covered by H¨ormander’s sum of the squares theorem but which can be seen to be globally hypoelliptic by the techniques described in this paper. For example, if X is a left-invariant vector field on S3 and ∂X is the derivative with respect to X, then the operator ∂X + c is globally hypoelliptic if and only if ic 6∈ 21 Z. We note that this operator is not locally hypoelliptic, but the knowledge of the global symbol allows us to draw conclusions about its global properties. It is interesting to observe that the number theory plays a role in the global properties, in some way similar to the appearance of the Liouville numbers in the global properties of vector fields on the torus. For example, the failure of the D’Alembertian D23 − D21 − D22 to be globally hypoelliptic can be related to properties of the so-called triangular numbers, and an

¨ HORMANDER CLASS Ψm (G) ON COMPACT LIE GROUPS

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explicit counterexample to the global hypoellipticity can be consequently constructed based on these numbers. We also look at the cases of the sub-Laplacian and other operators, which are covered by H¨ormander’s theorem, but for which we construct an explicit inverse which turns out to be a pseudo-differential operator with the global matrix-valued symbol of type ( 21 , 0). Such examples can also be extended to variable coefficient versions. Another application of Theorem 2.2 to obtain the sharp G˚ arding inequality on Lie groups will appear elsewhere. 1.2. We will now introduce some notation. Let throughout this paper G be a compact Lie group of (real) dimension n = dim G with the unit element e, and denote b the set of all equivalence classes of continuous irreducible unitary representaby G tions of G. For necessary details on Lie groups and their representations we refer to [7], but recall some basic facts for the convenience of the reader and in order to fix b corresponds to a homomorphism ξ : G → U(dξ ) with the notation. Each [ξ] ∈ G ξ(xy) = ξ(x)ξ(y) satisfying the irreducibility condition Cdξ = span{ξ(x)v : x ∈ G} for any given v ∈ Cdξ \ {0}. The number dξ is referred to as the dimension of the representation ξ. We always understand the group G as a manifold endowed with the (normalised) bi-invariant Riemannian structure. Of major interest for us is the following version b of the Peter-Weyl theorem defining the group Fourier transform F : L2 (G) → ℓ2 (G).

Theorem 1.1. The space L2 (G) decomposes as the orthogonal direct sum of bib invariant subspaces parameterised by G, M L2 (G) = Hξ , Hξ = {x 7→ Tr(Aξ(x)) : A ∈ Cdξ ×dξ }, b [ξ]∈G

the decomposition given by the Fourier series Z X b b f (x) = dξ Tr(ξ(x)f (ξ)), f (ξ) = f (x)ξ ∗ (x)dx. G

b [ξ]∈G

Furthermore, the following Parseval identity is valid, X kf k2L2 (G) = dξ kfb(ξ)k2HS . b [ξ]∈G

The notion of Fourier series extends naturally to C ∞ (G) and to the space of distributions D ′ (G) with convergence in the respective topologies. Any operator A on G mapping C ∞ (G) to D ′ (G) gives rise to a matrix-valued full symbol σA (x, ξ) ∈ Cdξ ×dξ σA (x, ξ) = ξ ∗ (x)(Aξ)(x),

so that (1)

Af (x) =

X

b [ξ]∈G

dξ Tr ξ(x) σA (x, ξ) fb(ξ)




holds as D ′ -convergent series. For such operators we will also write A = Op(σA ). For a rather comprehensive treatment of this quantisation we refer to [7] and [9]. We

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MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

denote the right-convolution kernel of A by RA , so that Z Z Af (x) = KA (x, y) f (y) dy = f (y) RA (x, y −1 x) dy, G

G

where dy is the normalised Haar measure on RG. The symbol σA and the rightconvolution kernel RA are related by σA (x, ξ) = G RA (x, y) ξ ∗ (y) dy. The first aim of this paper is to characterise H¨ormander’s class of pseudo-differential operators Ψm (G) by the behaviour of their full symbols, with applications to the ellipticity and global hypoellipticity of operators. The plan of this note is as follows. In the next section we give several equivalent symbolic characterisations of pseudo-differential operators by their symbols. These characterisations are based on difference operators acting on sequences of matrices b ∋ [ξ] 7→ σ(ξ) ∈ Cdξ ×dξ of varying dimension. We also give an application to G pseudo-differential operators on the group SU(2). Section 3 contains the proof and some useful lemmata on difference operators. Section 4 contains the criteria for the ellipticity of operators in terms of their full matrix-valued symbols as well as a finite version of the Leibniz formula for difference operators associated to the group representations. Finally, Section 5 contains criteria for the global hypoellipticity of pseudo-differential operators and a number of examples. ¨ rmander’s class 2. Characterisations of Ho The following statement is based on difference operators. They are defined as follows. We say that Qξ is a difference operator of order k if it is given by Qξ fb(ξ) = qd Q f (ξ)

for a function q = qQ ∈ C ∞ (G) vanishing of order k at the identity e ∈ G, i.e., (Px qQ )(e) = 0 for all left-invariant differential operators Px ∈ Diff k−1 (G) of order b k − 1. We denote the set of all difference operators of order k as diff k (G).

Definition 2.1. A collection of m ≥ n first order difference operators △1 , . . . , △m ∈ b is called admissible, if the corresponding functions q1 , . . . , qm ∈ C ∞ (G) satisfy diff 1 (G) dqj (e) 6= 0, j = 1, . . . , m, and rank(dq1 (e), . . . , dqm (e)) = n. It follows, in particular, m that e is an isolated common T zero of the family {qj }j=1 . An admissible collection is called strongly admissible if j {x ∈ G : qj (x) = 0} = {e}.

For a given admissible selection of difference operators on a compact Lie group G we use multi-index notation △αξ = △α1 1 · · · △αmm and q α (x) = q1 (x)α1 · · · qm (x)αm . (α) Furthermore, there exist corresponding differential operators ∂x ∈ Diff |α| (G) such that Taylor’s formula X 1 (2) f (x) = q α (x) ∂x(α) f (e) + O(h(x)N ), h(x) → 0, α! |α|≤N −1

holds true for any smooth function f ∈ C ∞ (G) and with h(x) the geodesic distance (α) from x to the identity element e. An explicit construction of operators ∂x in terms of q α (x) can be found in [7, Section 10.6]. In addition to these differential operators (α) ∂x ∈ Diff |α| (G) we introduce operators ∂xα as follows. Let ∂xj ∈ Diff 1 (G), j =

¨ HORMANDER CLASS Ψm (G) ON COMPACT LIE GROUPS

5

1, . . . , n, be a collection of left invariant first order differential operators corresponding to some linearly independent family of the left-invariant vector fields on G. We denote (α) ∂xα = ∂xα11 · · · ∂xαnn . We note that in most estimates we can freely replace operators ∂x by ∂xα and the other way around since they can be clearly expressed in terms of each other. After fixing the notion of difference operators, we have to specify orders. Each of the bi-invariant subspaces Hξ of Theorem 1.1 is an eigenspace of the Laplacian L on G with the corresponding eigenvalue −λ2ξ . Based on these eigenvalues we define hξi = (1 + λ2ξ )1/2 . In particular we recover the familiar characterisation f ∈ H s (G)

⇐⇒

b hξisfb(ξ) ∈ ℓ2 (G).

We are now in a position to formulate the main result of this paper. Theorem 2.2. Let A be a linear continuous operator from C ∞ (G) to D ′ (G). Then the following statements are equivalent: (A) A ∈ Ψm (G). (B) For every left-invariant differential operator Px ∈ Diff k (G) of order k and b of order ℓ the symbol estimate every difference operator Qξ ∈ diff ℓ (G) kQξ Px σA (x, ξ)kop ≤ CQξ Px hξim−ℓ

is valid. b we have (C) For any admissible selection △1 , . . . , △n ∈ diff 1 (G) k△αξ ∂xβ σA (x, ξ)kop ≤ Cαβ hξim−|α|

for all multi-indices α, β ∈ Nn0 . Moreover, sing supp RA (x, ·) ⊆ {e}. b we have (D) For a strongly admissible selection △1 , . . . , △n ∈ diff 1 (G) k△αξ ∂xβ σA (x, ξ)kop ≤ Cαβ hξim−|α|

for all multi-indices α, β ∈ Nn0 .

The set of symbols σA satisfying either of conditions (B)–(D) will be denoted by S m (G). Among other things, this theorem removes the assumption of the conjugation invariance from the list of conditions used in [7]. For u ∈ G, denote Au f := A(f ◦ φ) ◦ φ−1 , where φ(x) = xu. It can be shown that RAu (x, y) = RA (xu−1 , uyu−1) and σAu (x, ξ) = ξ(u)∗ σA (xu−1 , ξ) ξ(u).

On one hand, we have A ∈ Ψm (G) if and only if Au ∈ Ψm (G) for all u ∈ G. On the other hand, as a corollary of Theorem 2.2 this can be expressed in terms of difference operators, showing that the difference conditions are conjugation invariant. In fact, to draw such a conclusion, we even do not need to know that conditions (B)–(D) characterise the class Ψm (G): Corollary 2.3. Let the symbol σA of a linear continuous operator from C ∞ (G) to D ′ (G) satisfy either of the assumptions (B), (C), (D) of Theorem 2.2. Then the symbols σAu satisfy the same symbolic estimates for all u ∈ G.

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MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

This statement follows immediately from the condition (B) if we observe that the difference operators applied to the symbol σAu just lead to a different set of difference operators in the symbolic inequalities. Alternatively, one can observe that the change of variables by φ amounts to the change of the basis in the representation spaces of b thus leaving the condition (B) invariant again. G, Example 2.4. On the torus Tn = Rn /Zn , the family of functions qj (x) = e2πixj − 1, j = 1, . . . , n, gives rise to a strongly admissible collection of difference operators △j a(ξ) = △qj a(ξ) = a(ξ + ej ) − a(ξ),

j = 1, . . . , n,

cn ≃ Zn , where ej is the j th unit vector in Zn . As an immediate consequence with ξ ∈ T of Theorem 2.2 we recover the fact that the class Ψm (Tn ) can be characterised by the difference conditions on their toroidal symbols: |△αξ∂xβ σA (x, ξ)| ≤ Cαβ (1 + |ξ|)m−|α|

uniformly for all x ∈ Tn and ξ ∈ Zn . We refer to [8] for details of the corresponding toroidal quantisation of operators on Tn . See also [1]. In Theorem 2.2 we characterise H¨ormander operators entirely by symbol assumptions based on a set of admissible difference operators. There does not seem to be a canonic choice for difference operators, for different applications different selections of them seem to be most appropriate. We will comment on some of them. (1) Simplicity. Difference operators have a simple structure, if the corresponding functions are just matrix coefficients of irreducible representations. Then application of difference operators at fixed ξ involves only matrix entries from finitely many (neighbouring) representations. (2) Taylor’s formula. Any admissible selection of first order differences allows for a Taylor series expansion. (3) Leibniz rule. Leibniz rules have a particularly simple structure if one chooses difference operators of a certain form, see Proposition 4.3. Moreover, operators of this form yield a strongly admissible collection, see Lemma 4.4. Example 2.5. We will conclude this section with an application for the particular group SU(2). Certain explicit calculations on SU(2) have been done in [9] and the background on the necessary representation theory of SU(2) can be also found in [7]. Representations on SU(2) are parametrised by half-integers ℓ ∈ 21 N (so-called quantum numbers) and are of dimension dℓ = 2ℓ+1. Thus, irreducible representations of SU(2) are given by matrices tℓ (x) ∈ C(2ℓ+1)×(2ℓ+1) , ℓ ∈ 21 N, after some choice of the basis in the representation spaces. One admissible selection of difference operators corresponds to functions q+ , q0 , q0 ∈ C ∞ (SU(2)) defined by 1/2

q− = t−1/2,+1/2 ,

1/2

q+ = t+1/2,−1/2 ,

where we denote 1/2

t1/2 =

1/2

1/2

q0 = t−1/2,−1/2 − t+1/2,+1/2 , 1/2

t−1/2,−1/2 t−1/2,+1/2 1/2 1/2 t+1/2,−1/2 t+1/2,+1/2

!

.

Our analysis on SU(2) is in fact equivalent to the corresponding analysis on the 3sphere S3 ∼ = SU(2), the isomorphism given by the identification of SU(2) with S3 ⊂ H

¨ HORMANDER CLASS Ψm (G) ON COMPACT LIE GROUPS

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in the quaternion space H, with the quaternionic product on S3 corresponding to the matrix multiplication on SU(2). Writing explicitly an isomorphism Φ : S3 → SU(2) as   x0 + ix3 x1 + ix2 (x0 , x1 , x2 , x3 ) = x 7→ Φ(x) = , −x1 + ix2 x0 − ix3 the identity matrix e ∈ SU(2) corresponds to the vector 1 in the basis decomposition x = (xm )3m=0 7→ x0 1 + x1 i + x2 j + x3 k on H. We note that the family △+ := △q+ , △− := △q− and △0 := △q0 is not strongly admissible because in addition to e ∈ SU(2) (or to 1 ∈ H) they have another common zero at −e (or at −1). Following [7] and [9], we simplify the notation on SU(2) and S3 by writing σA (x, ℓ) for σA (x, tℓ ), ℓ ∈ 12 N, and we refer to these works for the explicit formulae for the difference operators △+ , △− , △0. We denote △αℓ = △α+1 △α−2 △α0 3 . In [9] we proved that the operator and the Hilbert-Schmidt norms are uniformly equivalent for symbols of pseudo-differential operators from Ψm (SU(2)). As a corollary to Theorem 2.2, Corollary 2.3, and [7, Theorem 12.4.3] we get Theorem 2.6. Let A be a linear continuous operator from C ∞ (SU(2)) to D ′ (SU(2)). Then A ∈ Ψm (SU(2)) if and only if sing supp RA (x, ·) ⊂ {e} and k△αℓ ∂xβ σA (x, ℓ)kHS ≤ Cαβ (1 + ℓ)m−|α| for all multi-indices α, β ∈ N3 and all l ∈ 21 N. Moreover, in these cases we also have the rapid off-diagonal decay property of symbols, namely, we have α β △ ∂ σA (x, ℓ)ij ≤ CAαβmN (1 + |i − j|)−N (1 + ℓ)m−|α| ℓ x

uniformly in x ∈ SU(2), for every N ≥ 0, all ℓ ∈ 21 N, every multi-indices α, β ∈ N30 , and for all matrix column/row numbers i, j.

We note that Theorem 2.6 holds in exactly the same way if we replace SU(2) by S3 . Moreover, according to Theorem 1.1 the statement of Theorem 2.6 holds without the kernel condition sing supp RA (x, ·) ⊂ {e} provided that instead of the admissible family △+ , △− , △0 we take a strongly admissible family of difference operators in \ diff 1 (SU(2)). 3. Proof of Theorem 2.2 3.1. (A) =⇒ (C). By [7, Thm. 10.9.6] (and in the used there) we know T notation m m that condition (A) is equivalent to σA ∈ Σ (G) = k Σk (G), where Σm 0 (G) already corresponds to assumption (C). There is nothing to prove. 3.2. (C) =⇒ (D). Evident. Also the following partial converse is true. If (D) is satisfied, then the symbol estimates imply for the corresponding right-convolution kernel RA (x, ·) of A that sing supp RA (x, ·) ⊆

n \

{y ∈ G : qj (y) = 0} = {e}.

j=1

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MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

3.3. (D) =⇒ (B). For a given strongly admissible selection of first order differences, we can apply Taylor’s formula to the function qQ corresponding to the difference b Hence, we obtain operator Q ∈ diff ℓ (G). b be an arbitrary difference operator of order ℓ. Then Lemma 3.1. Let Q ∈ diff ℓ (G) X Q= cα △αξ + QN ℓ≤|α|≤N −1

b for suitable constants cα ∈ C and a difference operator QN ∈ diff N (G).

Consequently, (D) together with (B) for differences of order larger than N implies (B). More precisely, with the notation kσ(x, ξ)kC ∞ for ‘any’ C ∞ -seminorm of the form supx kPx σ(x, ·)kop and m RN = {σA : kQξ σA (x, ξ)kC ∞ . hξim−ℓ

and we obtain

b ∀ℓ > N ∀Qξ ∈ diff ℓ (G)}

Sℓm = {σA : k△αξσA (x, ξ)kC ∞ . hξim−ℓ

∀|α| = ℓ},

m R0m = S0m ∩ S1m ∩ · · · ∩ SNm ∩ RN ,

while (B) corresponds to taking all symbols from R0m . This implies ! \ m Sℓm ∩ R∞ R0m = ℓ≥0

with

b ∀N ∀Qξ ∈ diff ∞ (G)}. T Assumption (D) corresponds to the first intersection S m = ℓ Sℓm . It remains to m show that S m ⊂ R∞ , which is done in the sequel. ∞ b ∞ Let QP ξ ∈ diff (G), i.e. qQ ∈ C (G) is vanishing to infinite order at e. Let further n 2 p(y) = j=1 qj (y). Then p(y) 6= 0 for y 6= e and p vanishes of the second order at y = e. Hence, qQ (y)/|p(y)|N ∈ C ∞ (G) for arbitrary N and, therefore, m R∞ = {σA : kQξ σ(x, ξ)kC ∞ . CN hξi−N

qQ (y)RA (x, y) =

qQ (y) |p(y)|N RA (x, y) |p(y)|N

can be estimated in terms of symbol bounds for σA ∈ S m . Indeed, by Lemma 3.2 it follows that |p(y)|N RA (x, y) ∈ H −m−c+2N (G × G). Letting N → ∞ proves the claim.

Lemma 3.2. Let σA ∈ S m . Then RA (x, ·) ∈ H −m−c (G) for all c > 21 dim G. p Proof. This just follows from kσA (x, ξ)kHS ≤ dξ kσA (x, ξ)kop in combination with the observation that X d2ξ hξi−2c < ∞ [ξ]

for c >

1 2

dim G (which is equivalent to H c (G) ֒→ L2 (G) being Hilbert-Schmidt). 

3.4. (B) ⇐⇒ (C) ⇐⇒ (D). Evident from the above.

¨ HORMANDER CLASS Ψm (G) ON COMPACT LIE GROUPS

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3.5. (B) =⇒ (A). We use the commutator characterisation of the class Ψm (G), [7, Thm. 10.7.7]. It says that it is sufficient to show that (1) any symbol σA satisfying condition (B) gives rise to a bounded linear operator A : H m (G) → L2 (G); (2) the symbol σ[X,A] of the commutator of A and any left-invariant vector field X also satisfies (B). The nuclearity of C ∞ (G) allows one to split the first statement into the corresponding results for purely x- and purely ξ-dependent symbols; both situations are evident. As for the second statement, we use the symbolic calculus established in [7, Thm. 10.7.9] for classes of symbols including, in particular, the class S m , which writes the symbol of the composition as the asymptotic sum σ[X,A] ∼ σX σA − σA σX +

X 1 (△α σX )(∂x(α) σA ). α! ξ

|α|>1

The first term σX σA − σA σX is on the level of right-convolution kernels RA (x, ·) given by (Xy − XyR )RA (x, y), where X R is the right-invariant vector field tangent to X at e (and the index y describes the action as a differential operator with respect to the y-variable). Apparently, Xy − XyR is a smooth vector field on G and can thus be written as n X

qj (y)∂jR

j=1

with qj (e) = 0 and ∂jR suitable right-invariant derivatives. Hence, on the symbolic Pn 1 b b level σX σA − σA σX = j=1 △j (σA σ∂j ) with △j ∈ diff (G), defined by △j f (ξ) = qc j f (ξ). Now the statement follows from Lemma 3.4 by the aid of the asymptotic Leibniz rule [7, Thm. 10.7.12]: b be a set of admissible differences. For any Lemma 3.3. Let △1 , . . . , △n ∈ diff 1 (G) ℓ b ℓ b Qξ ∈ diff (G) there exist Qξ,α ∈ diff (G) such that X Qξ (σ1 σ2 ) ∼ (Qξ σ1 )σ2 + (Qξ,α σ1 )(△αξ σ2 ) |α|>0

for any (fixed) set of admissible differences. Lemma 3.4. Let A be a differential operator of order m. Then σA ∈ S m . Proof. Lemma 3.4 is a consequence of the already proved implications (A) =⇒ (C) =⇒ (D) =⇒ (B). Explicit formulae for the difference operators applied to symbols of differential operators can be also found in [7, Prop. 10.7.4] and provide an elementary proof of this statement. 

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MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

4. Ellipticity and Leibniz formula As an application of Theorem 2.2 we will give a characterisation of the elliptic operators in Ψm (G) in terms of their global symbols. Theorem 4.1. An operator A ∈ Ψm (G) is elliptic if and only if its matrix valued b and for all such ξ symbol σA (x, ξ) is invertible for all but finitely many [ξ] ∈ G, satisfies (3)

for all x ∈ G.

kσA (x, ξ)−1 kop ≤ Chξi−m

Thus, both statements are equivalent to the existence of B ∈ Ψ−m (G) such that I − BA and I − AB are smoothing smoothing. For the ellipticity condition (3) on the general matrix level it is not enough to assume that | det σA (x, ξ)|1/dξ ≥ Chξim due to the in general growing dimension of the matrices. However, if we assume that the smallest singular value of the matrix σA (x, ξ) is greater or equal than Chξim uniformly in x and (all but finitely many) ξ, then condition (3) follows. Let the collection q1 , . . . , qn give an admissible collection of difference operators and (γ) let ∂x be the corresponding family of differential operators as in the Taylor expansion formula (2). As an immediate corollary of Theorem 4.1 and [7, Thm. 10.9.10] we get Corollary 4.2. Let A ∈ Ψm (G) be elliptic. Moreover, assume P∞ that σA (x, ξ) ∼ P ∞ m−j (G). Let σB (x, ξ) ∼ k=0 σBk (x, ξ), where j=0 σAj (x, ξ), where Op(σAj ) ∈ Ψ −1 σB0 (x, ξ) = σA0 (x, ξ) for large ξ, and the symbols σBk are defined recursively by (4)

σBN (x, ξ) = −σB0 (x, ξ)

N −1 N −k X X

X

k=0 j=0 |γ|=N −j−k

 1  γ △ξ σBk (x, ξ) ∂x(γ) σAj (x, ξ). γ!

Then Op(σBk ) ∈ Ψ−m−k (G), B = Op(σB ) ∈ Ψ−m (G), and the operators AB − I and BA − I are in Ψ−∞ (G). First, we give some preliminary results. In Lemma 3.3 we gave an asymptotic Leibniz formula but here wewill present its finite version. Given a continuous unitary matrix representation ξ = ξij 1≤i,j≤ℓ : G → Cℓ×ℓ , ℓ = dξ , let q(x) = ξ(x) − I (i.e. qij = ξij − δij with Kronecker’s deltas δij ), and define Dij fb(ξ) := qd ij f (ξ).

In the previous notation, we can write Dij = ∆qij . However, here, to emphasize that qij ’s correspond to the same representation, and to distinguish with difference operators with a single subindex, Pℓ we will use the notation D instead. For a multi-index ℓ2 γ ∈ N0 , we will write |γ| = i,j=1 |γij |, and for higher order difference operators we γℓ,ℓ−1 γℓℓ write Dγ = Dγ1111 Dγ1212 · · · Dℓ,ℓ−1 Dℓℓ . For simplicity, we will abbreviate writing a = a(ξ) and b = b(ξ). Let k ∈ N0 and let α, β ∈ Nk be such that 1 ≤ αj , βj ≤ ℓ for all j ∈ {1, · · · , k}. Let us define a grand →k − kth order difference D by →k − →k − ( D a)αβ = D αβ a := Dα1 β1 · · · Dαk βk a,

¨ HORMANDER CLASS Ψm (G) ON COMPACT LIE GROUPS

11

where we may note that the first order differences Dαj βj commute with each other. We may compute with “matrices” h i →k − →k − D a = D αβ a k α,β∈{1,··· ,ℓ}

using the natural operations, e.g.   →k − − →k ( D a)( D b)

αβ

:=

X

γ∈{1,··· ,ℓ}k



  −k → →k − D αγ a D γβ b .

In contrast to Lemma 3.3 with an asymptotic Leibniz formula for arbitrary difference →k − operators, operators D satisfy the finite Leibniz formula: Proposition 4.3. For all k ∈ N and α, β ∈ {1, · · · , ℓ}k we have X →k − k D αβ (ab) = Cαβεδ (Dε a) (Dδ b), |ε|,|δ|≤k≤|ε|+|δ|

2

with the summation taken over all ε, δ ∈ Nℓ0 satisfying |ε|, |δ| ≤ k ≤ |ε| + |δ|. In particular, for k = 1, we have − →1 − →1  Dij (ab) = (Dij a) b + a (Dij b) + ( D a)( D b) (5) ij

= (Dij a) b + a (Dij b) +

n X

(Dik a) (Dkj b) .

k=1

Proof. Let us first show the case of k = 1 given in formula (5). Recalling the notation qij = ξij − δij , from the unitarity of the representation ξ, for all x, y ∈ G, we readily obtain the identity n X −1 qik (xy −1 )qkj (y). (6) qij (x) = qij (xy ) + qij (y) + k=1

∞

If now f, g ∈ C (G), we consequently get

qij · (f ∗ g) = (qij f ) ∗ g + f ∗ (qij g) +

n X k=1

(qik f ) ∗ (qkj g),

which on the Fourier transform side gives (5). For k ≥ 2, let us argue inductively. Taking differences is linear, so the main point is to analyse the application of a first order difference Dα1 β1 to the term − →1 − →1  ( D a)( D b) . By (5) we have α2 β2 − −  − →1 − →1  →1  − →1 →1 − →1 = ( D Dα1 β1 a)( D b) Dα1 β1 ( D a)( D b) + ( D a)( D Dα1 β1 b) α2 β2 α2 β2 α2 β2 − →2 − →2  + ( D a)( D b) , αβ

where α = (α1 , α2 ) and β = (β1 , β2 ). Proceeding by induction, we notice that   →k − − →k applying a first order difference to the term ( D a)( D b) would introduce a αβ

12

MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

sum of old type terms plus the new term



− k+1 − → →k+1 ( D a)( D b)



. Thus, a kth order

α eβe

difference operator applied to the product symbol ab produces a linear combination of terms of the form (Dε a)(Dδ b), where |ε|, |δ| ≤ k ≤ |ε| + |δ|, completing the proof.  Difference operators D associated to the representations may be particularly useful in view of the finite Leibniz formula in Proposition 4.3 and the fact that the collection of all differences Dij over all ξ and i, j is strongly admissible: Lemma 4.4. The family of difference operators associated to the family of functions {qij = ξij − δij }[ξ]∈G, b 1≤i,j≤dξ is strongly admissible. Moreover, this family has a finite subfamily associated to finitely many representations which is still strongly admissible. Proof. We observe that there exists a homomorphic embedding of G into U(N) for a large enough N and this embedding itself is a representation of G of dimension N. Decomposing this representation into irreducible components gives a finite collection of representations. The common zero set of the corresponding family {qij } is e which means that it is strongly admissible.  We note that on the group SU(2) or on S3 , by this argument, or by the discussion \ in Section 2 the four function qij corresponding to the representation [tℓ ] ∈ SU(2) with ℓ = 21 of dimension two, dℓ = 2, already give a strongly admissible collection of four difference operators. We can now apply Proposition 4.3 to the question of b We formulate this for symbol classes with (ρ, δ) inverting the symbols on G × G. behaviour as this will be also used in Section 5.

Lemma 4.5. Let m ≥ m0 , 1 ≥ ρ > δ ≥ 0 and let us fix difference operators b Let the matrix symbol {Dij }1≤i,j≤dξ0 corresponding to some representation ξ0 ∈ G. a = a(x, ξ) satisfy (7)

kDγ ∂xβ a(x, ξ)kop ≤ Cβγ hξim−ρ|γ|+δ|β|

for all multi-indices β, γ. Assume also that a(x, ξ) is invertible for all x ∈ G and b and satisfies [ξ] ∈ G, ka(x, ξ)−1 kop ≤ Chξi−m0

(8)

b and if m0 6= m in addition that for all x ∈ G and [ξ] ∈ G,

 

a(x, ξ)−1 Dγ ∂ β a(x, ξ) ≤ Chξi−ρ|γ|+δ|β| (9) x op

b Then the matrix symbol a−1 defined by a−1 (x, ξ) = a(x, ξ)−1 for all x ∈ G and [ξ] ∈ G. satisfies (10)

kDγ ∂xβ a−1 (x, ξ)kop ≤ Cβγ hξi−m0 −ρ|γ|+δ|β|

for all multi-indices β, γ. Proof. Let us denote b(x, ξ) := a(x, ξ)−1 and estimate ∂xβ b first. Suppose we have proved that (11)

k∂xβ b(x, ξ)kop ≤ Cβ hξi−m0 +δ|β|

¨ HORMANDER CLASS Ψm (G) ON COMPACT LIE GROUPS

13

whenever |β| ≤ k. We proceed by induction. Let us study the order k + 1 cases ∂xj ∂xβ b with |β| = k. Since a(x, ξ)b(x, ξ) = I, by the usual Leibniz formula we get X a ∂xj ∂xβ b = − Cβ1 β2 (∂xβ1 a) (∂xβ2 b). β1 +β2 =β+ej ,|β2 |≤|β|

From (7), (8), (9) and (11) we obtain the desired estimate k∂xj ∂xβ b(x, ξ)kop ≤ Cβ hξi−m0 +δ|β|+δ .

Let us now estimate Dγ b. The argument is more complicated than that for the ∂x derivatives because the Leibniz formula in Proposition 4.3 has more terms. Indeed, the Leibniz formula in Proposition 4.3 applied to a(x, ξ)b(x, ξ) = I gives Dij b + b(Dij a)b +

dξ0 X

b(Dik a)Dkj b = 0.

k=1

Writing these equations for all 1 ≤ i, j ≤ dξ0 gives a linear system on {Dij b}ij with coefficients of the form I + bDkl a for suitable sets of indices k, l. Since kbDkl akop ≤ Chξi−ρ , we can solve it for {Dij b}ij for large hξi. The inverse of this system is bounded, and kb(Dij a)bkop ≤ Chξi−m0 −ρ , implying that kDij bkop ≤ Chξi−m0 −ρ . Suppose now we have proved that (12)

kDγ b(x, ξ)kop ≤ Cγ hξi−m0 −ρ|γ|

whenever |γ| ≤ k. We proceed by induction. Let us study the order k + 1 cases Dij Dγ b with |γ| = k. Now a(x, ξ)b(x, ξ) = I, so that Dij Dγ (ab) = 0,

where by Proposition 4.3 the right hand side is a sum of terms of the form (Dε a) (Dη b) , where |ε|, |η| ≤ k + 1 ≤ |ε| + |η|. Especially, we look at the terms Dε a Dη b with |η| = k + 1, since the other terms can be estimated by hξi−ρ(k+1) due to (12) and (7). ′ Writing the linear system of equations on Dij Dη b produced by the Leibniz formula for all i, j and all |η ′| = k, we see that the matrix coefficients in front of matrices ′ Dij Dη b are sums of the terms of the form Dε a. The main term in each coefficient is a(x, ξ) corresponding to ε = 0, while the other terms corresponding to ε 6= 0 can be ′ estimated by hξim−ρ in view of (12) and (7). It follows that we can solve Dij Dη b from this system and the solution matrix can be estimated by hξi−m0 in view of (8) since ′ its main term is a(x, ξ). Therefore, all terms of the type Dij Dη can be estimated by



η′

Dij D b(x, ξ) ≤ Cη′ ,i,j hξi−m0 −ρ(k+1) , op

which is what was required to prove. By combining these two arguments we obtain estimate (10). 

Proof of Theorem 4.1. Assume first that A ∈ Ψm (G) is elliptic. Then it has a parametrix B ∈ Ψ−m (G) such that AB − I, BA − I ∈ Ψ−∞ (G). By the composition formulae for the matrix valued symbols in [7, Thm. 10.7.9] and Theorem 2.2 it follows that Op(σA σB ) − I ∈ Ψ−1 (G). Consequently, the product σA (x, ξ)σB (x, ξ) is an invertible matrix for all sufficiently large hξi and k(σA σB )−1 kop ≤ C. It follows

14

MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

then also that σA (x, ξ) is an invertible matrix for all sufficiently large hξi. From this and the equality σA−1 (σA σB ) = σB we obtain kσA−1 kop = kσB (σA σB )−1 kop ≤ kσB kop k(σA σB )−1 kop ≤ Chξi−m .

Assume now that A ∈ Ψm (G) has the invertible matrix symbol σA satisfying kσA (x, ξ)−1 kop ≤ Chξi−m . We can disregard representations ξ with bounded hξi because they correspond to a smoothing operator. From Theorem 2.2 applied with the difference operators from Lemma 4.4, we get that Op(σA−1 ) ∈ Ψ−m (G) because we have kDγ ∂xβ (σA (x, ξ)−1 )kop ≤ Chξi−m−|γ| by Lemma 4.5 with m0 = m, ρ = 1 and δ = 0. Since Op(σA−1 ) ∈ Ψ−m (G), Theorem 10.9.10 in [7] implies that A has a parametrix (given by formula (4)). This implies that A is elliptic. The proof is complete.  5. Global hypoellipticity We now turn to the analysis of hypoelliptic operators. We recall that an operator A is globally hypoelliptic on G if Au = f and f ∈ C ∞ (G) imply u ∈ C ∞ (G). We m use the notation Sρ,δ (G) for the class of symbols for which the corresponding (ρ, δ)m versions of symbol estimates are satisfied, namely, σA ∈ Sρ,δ (G) if for a strongly 1 b admissible selection △1 , . . . , △n ∈ diff (G) we have k△αξ ∂xβ σA (x, ξ)kop ≤ Cαβ hξim−ρ|α|+δ|β|

for all multi-indices α, β ∈ Nn0 . Theorem 2.2 (B) ⇔ (C) ⇔ (D) is valid for all m 1 ≥ ρ > δ ≥ 0. By Op(Sρ,δ (G)) we denote the class of all operators A of the m (G). Such operators can be readily seen to be form (1) with symbols σA ∈ Sρ,δ ∞ m continuous on C (G). In the previous notation we have S m (G) = S1,0 (G), and m m Ψ (G) = Op(S1,0 (G)) by Theorem 2.2.

5.1. The knowledge of the full symbols allows us to establish an analogue of the well-known hypoellipticity result of H¨ormander [5] on Rn , requiring conditions on lower order terms of the symbol. The following theorem is a matrix-valued symbol criterion for (local) hypoellipticity. m (G)) be a Theorem 5.1. Let m ≥ m0 and 1 ≥ ρ > δ ≥ 0. Let A ∈ Op(Sρ,δ m pseudo-differential operator with the matrix-valued symbol σA (x, ξ) ∈ Sρ,δ (G) which b and for all such ξ satisfies is invertible for all but finitely many [ξ] ∈ G,

(13)

kσA (x, ξ)−1kop ≤ Chξi−m0

for all x ∈ G. Assume also that for a strongly admissible collection of difference operators we have

 

σA (x, ξ)−1 △αξ ∂xβ σA (x, ξ) ≤ Chξi−ρ|α|+δ|β| (14) op

for all multi-indices α, β, all x ∈ G, and all but finitely many [ξ]. Then there exists −m0 an operator B ∈ Op(Sρ,δ (G)) such that AB − I and BA − I belong to Ψ−∞ (G). Consequently, we have sing supp Au = sing supp u for all u ∈ D ′ (G).

¨ HORMANDER CLASS Ψm (G) ON COMPACT LIE GROUPS

15

Proof. First of all, we can assume at any stage of the proof that (13) and (14) hold for b since we can always modify symbols for small hξi which amounts to adding all [ξ] ∈ G a smoothing operator. Then we observe that if we apply Theorem 2.2 with difference −m0 operators D from Lemma 4.4, it follows from Lemma 4.5 that σA−1 ∈ Sρ,δ (G). Let −ρ|α|+δ|β|

us show next that σA−1 △αξ ∂xβ σA ∈ Sρ,δ (15) we get

(G). Differentiating the equality

σA (σA−1 △αξ ∂xβ σA ) = △αξ ∂xβ σA ,


∂xj σA−1 △αξ ∂xβ σA = σA−1 ∂xj △αξ ∂xβ σA − (σA−1 ∂xj σA )(σA−1 △αξ ∂xβ σA ).
From this and (14) it follows that k∂xj σA−1 △αξ ∂xβ σA kop ≤ Chξi−ρ|α|+δ|β|+δ . Continuing
this argument, we get that k∂xγ σA−1 △αξ ∂xβ σA kop ≤ Chξi−ρ|α|+δ|β+γ| for all γ. For differences, the Leibniz formula in Proposition 4.3 applied to (15) gives Dij (σA−1 △αξ ∂xβ σA ) + (σA−1 Dij σA )(σA−1 △αξ ∂xβ σA ) X + (σA−1 Dik σA )Dkj (σA−1 △αξ ∂xβ σA ) = σA−1 Dij △αξ ∂xβ σA . k

Writing these equations for all i, j gives a linear system on {Dij (σA−1 △αξ ∂xβ σA )}ij with coefficients of the form I + σA−1 Dkl σA for suitable sets of indices k, l. Since kσA−1 Dkl σA kop ≤ Chξi−ρ by (14) and Theorem 2.2, we can solve it for matrices {Dij (σA−1 △αξ ∂xβ σA )}ij for large hξi. Finally, since k(σA−1 Dij σA )(σA−1 △αξ ∂xβ σA )kop ≤ Chξi−ρ|α|+δ|β|−ρ, kσA−1 Dij △αξ ∂xβ σA kop ≤ Chξi−ρ|α|+δ|β|−ρ ,
we get that kDij σA−1 △αξ ∂xβ σA kop ≤ Chξi−ρ|α|+δ|β|−ρ for all i, j. An induction argument similar to the one in the proof of Lemma 4.5 shows that σA−1 △αξ ∂xβ σA ∈ P −ρ|α|+δ|β| Sρ,δ (G). Let us now denote σB0 (x, ξ) = σA0 (x, ξ)−1 , and let σB ∼ ∞ N =0 σBN , where N −1 X X  1  γ △ξ σA (x, ξ) ∂x(γ) σBk (x, ξ). (16) σBN (x, ξ) = −σB0 (x, ξ) γ! k=0 |γ|=N −k

One can readily check that B is a parametrix of A (this is a special case of [7, Exercise −m0 −m0 −N (G). We already know that σB0 ∈ Sρ,δ (G). 10.9.12]). We claim that σBN ∈ Sρ,δ −ρ|α|+δ|β| −1 α β Inductively, in view of σA △ξ ∂x σA ∈ Sρ,δ (G), formula (16) implies that the order of BN is −(ρ−δ)|γ|+order(Bk ) = −(ρ−δ)|γ|−m0 −(ρ−δ)k = −m0 −(ρ−δ)N. −m0 Thus, B ∈ Op(Sρ,δ (G)) is a parametrix for A. Finally, the pseudo-locality of the operator A (which can be proved directly from the symbolic calculus) implies that sing supp Au ⊂ sing supp u. Writing u = B(Au)− Ru with R ∈ Ψ−∞ (G), we get that [ sing supp u ⊂ sing supp(Au) sing supp(Ru), so that sing supp Au = sing supp u because Ru ∈ C ∞ (G).



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MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

To obtain global hypoellipiticity, it is sufficient to show that an operator A ∈ Ψ(G) has a parametrix B satisfying subelliptic estimates kBf kH s . kf kH s+m for some constant m independent of s ∈ R. 5.2. Examples. We will conclude with a collection of examples on the group SU(2). These examples can be also viewed as examples on the 3-sphere S3 , see Example 2.5. Thus, in what follows we denote G = S3 ∼ = SU(2). In contrast to [7] we will base considerations here on the four difference operators Dij , i, j ∈ {1, 2}, defined in terms of the first non-trivial representation corresponding to ℓ = 21 . They are related to △η , η ∈ {0, +, −} (see Example 2.5), via △− = D12 , △+ = D21 , and △0 = D11 −D22 , while b and satisfy the Leibniz rule (5). Furthermore, the elementary D11 + D22 ∈ diff 2 (G), (homogeneous) first order differential operators ∂0 , ∂± ∈ Ψ1 (G) and the Laplacian L = −∂02 − 21 (∂− ∂+ + ∂+ ∂− ) ∈ Ψ2 (G) satisfy σ∂0

σ∂+

σ∂−

σL

D11

1 σ 2 I

0

0

−σ∂0 + 14 σI

D12

0

σI

0

−σ∂−

D21

0

0

σI

−σ∂+

D22

− 12 σI

0

0

σ∂0 + 14 σI

Invariant vector fields corresponding to the basis Pauli matrices in su(2) are expressible as D1 = − 2i (∂− + ∂+ ), D2 = 21 (∂− − ∂+ ) and D3 = −i∂0 . We refer to [7, 9] for the details on these operators and for the explicit formulae for their symbols. Example 5.2. We consider D3 + c for constants c ∈ C. The corresponding symbol is given by ℓ (σD3 +c )ℓmn = [c − im]δmn

where, following [7], m, n ∈ {−ℓ, −ℓ + 1, . . . , ℓ − 1, ℓ} run through half-integers or integers, depending on whether ℓ is a half-integer or an integer, respectively. We also ℓ ℓ ℓ use the notation δmn for the Kronecker delta, i.e. δmn = 1 for m = n and δmn = 0 for m 6= n. This symbol σD3 +c is invertible for all ℓ if (and only if) c is not an imaginary 0 half-integer, ic 6∈ 21 Z. The inverse satisfies (D3 + c)−1 ∈ Op(S0,0 (G)), based on D11 σ(D3 +c)−1 = − 21 σ(D3 +c)−1 σ(D3 +c+ 1 )−1 , 2

D22 σ(D3 +c)−1 =

1 −1 σ σ (D3 +c− 21 )−1 , 2 (D3 +c)

and D12 σ(D3 +c)−1 = D21 σ(D3 +c)−1 = 0, combined with Leibniz’ rule. As a consequence we see that the operator D3 + c satisfies sub-elliptic estimates with loss of one derivative and are thus globally hypoelliptic. The statement is sharp: the spectrum of D3 consists of all imaginary half-integers and all eigenspaces are infinite-dimensional (stemming from the fact that each such imaginary half-integer hits infinitely many representations for which −ℓ ≤ ic ≤ ℓ and ic + ℓ ∈ Z). In particular, eigenfunctions can be irregular, e.g., distributions which do not belong to certain negative order Sobolev spaces.

¨ HORMANDER CLASS Ψm (G) ON COMPACT LIE GROUPS

17

Example 5.3. By conjugation formulae in [9, 7] this example can be extended to give examples of more general left-invariant operators. Namely, let X ∈ g and let ∂X be the derivative with respect to the vector field X. Then the operator ∂X + c is globally hypoelliptic if and only if ic 6∈ 21 Z. By conjugation again, we can readily see that in a suitably chosen basis in the k representation spaces, the symbol of the operator ∂X , k ∈ N, is ℓ (σ∂Xk )ℓmn = [(−im)k ]δmn .

k Denoting ( 12 Z)k = {2−k lk : l ∈ Z}, we see that the operator ∂X + c is globally k k 1 hypoelliptic if and only if c 6∈ −(−i) ( 2 Z) . In particular, for k = 2, the second order 2 differential operator ∂X + c is globally hypoelliptic if and only if 4c 6∈ {l2 : l ∈ Z}. 2 It is also interesting to note that the operators ∂X + a(x)∂X are not globally hy∞ poelliptic for all complex-valued functions a ∈ C (SU(2)). Indeed, in a suitably \ their symbols are diagonal, and chosen basis of the representation spaces in SU(2) ℓ (σ∂X2 +a(x)∂X )mn = 0 for ℓ ∈ N and m = n = 0. For simplicity in the following construction we assume that ∂X = D3 . If we set fb(ℓ)mn = 1 for ℓ ∈ N and m = n = 0, and to be zero otherwise, we have (σ∂ 2 +a(x)∂ )fb = 0, so that (∂ 2 + a(x)∂X )f = 0 by X

X

X

(1). By the Fourier inversion formula we see that fb is the Fourier transform of the function X (2ℓ + 1)tℓ00 (x), f (x) = ℓ∈N0

′

2 so that f ∈ D (SU(2)) is not smooth but (∂X + a(x)∂X )f = 0. We note that operators from Example 5.3 are not covered by H¨ormander’s sum of the squares theorem [6]. Although the following example of the sub-Laplacian is covered by H¨ormander’s theorem, Theorem 5.1 provides its inverse as a pseudodifferential operator with the global matrix-valued symbol in the class S 1−1 (G). ,0 2

Example 5.4. Sub-Laplacian. The sub-Laplacian on SU(2) is defined as Ls = D21 + D22 ∈ Ψ2 (G) and has the symbol ℓ . (σLs )ℓmn = [m2 − ℓ(ℓ + 1)]δmn

This symbol is invertible for all ℓ > 0. It satisfies the hypoellipticity assumption of Theorem 5.1 with ρ = 12 . For this we note that D11 σLs = D22 σLs = 0 and D12 σLs = −σ∂− , D21 σLs = −σ∂+ . Estimate (14) is evident for α = 0 and |α| ≥ 2. Hence it remains to check that k(σLs )−1 D12 σLs k = k(σLs )−1 σ∂− k . ℓ−1/2

(and the analogous statement for D21 ). This can be done by direct computation, # "p (ℓ + n)(ℓ − n + 1) ℓ δm,n−1 , (σL−1s D12 σLs )ℓmn = (n − 1)2 − ℓ(ℓ + 1) p the entries corresponding to ℓ = −n vanish, while all others are bounded by 2/ℓ. (G)). In particular, we Hence, Ls has a pseudo-differential parametrix L♯s ∈ Op(S 1−1 ,0 2

automatically get the subelliptic estimate kf kH s(G) ≤ CkLs f kH s−1 (G) for all s ∈ R.

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MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

Example 5.5. Heat operator. We consider the analogue of the heat operator on SU(2), namely the operator H = D3 − D21 − D22 . Writing H = D3 − Ls , we see that its symbol is ℓ (σH )ℓmn = [−im − m2 + ℓ(ℓ + 1)]δmn .

Consequently, using the differences for the sub-Laplacian, we see that D11 σH = − 2i σI , D22 σH = 2i σI , D12 σH = −D12 σLs = σ∂− , D21 σH = −D21 σLs = σ∂+ . By an argument similar to that for the sub-Laplacian, the operator H has a parametrix H ♯ ∈ Op(S 1−1 (G)). ,0 2

Example 5.6. Schr¨odinger operator. We consider the analogue of the Schr¨odinger operator on SU(2), namely the operators S± = ±iD3 − D21 − D22 . We will treat both ± cases simultaneously by keeping the sign ± or ∓. Writing S± = ±iD3 − Ls , we see that their symbols are ℓ (σS± )ℓmn = [±m − m2 + ℓ(ℓ + 1)]δmn .

Consequently, we see that these symbols are not invertible. Analogously to Example 5.3 we can construct distributions in the null spaces of S± . Namely, if we define fc = n = ∓ℓ, and to be zero otherwise, then we readily see that the ± (ℓ)mn = 1 for m P functions f± (x) = ℓ∈ 1 N0 (2ℓ + 1)tℓ∓ℓ∓ℓ (x) satisfy S± f± = 0, but f± ∈ D ′ (G) are not 2 smooth. On the other hand, the operators S± + c become globally hypoelliptic for almost all constants c. Let us consider the case S− , the argument for S+ is similar. Since the symbol of S− + c is ℓ , (σS− +c )ℓmn = [c − m − m2 + ℓ(ℓ + 1)]δmn

we see that it is invertible if and only if c 6∈ {ℓ(ℓ + 1) − m(m + 1) : ℓ ∈ 21 N, ℓ − m ∈ Z, |m| ≤ ℓ}. If, for example, c > 0 or c ∈ C\R, then kσ(S− +c)−1 kop ≤ C, and hence S− + c is globally hypoelliptic. Example 5.7. D’Alembertian. We consider W = D23 − D21 − D22 . Writing W = D23 − Ls = 2D23 − L with the Laplace operator L = D21 + D22 + D23 , we see that the symbol of W ∈ Ψ2 (G) is given by ℓ (σW )ℓmn = [−2m2 + ℓ(ℓ + 1)]δmn .

If ℓ is a half-integer, we easily see that | − 2m2 + ℓ(ℓ + 1)| ≥ 41 for all ℓ − m ∈ Z, |m| ≤ ℓ. However, if ℓ is an integer, the triangular numbers ℓ(ℓ+1) are squares if 2 √ k ℓ = ℓk = ⌊(3+2 2) /4⌋, k ∈ N, see [14, A001109]. Denoting by mk the corresponding solution to 2m2k = ℓk (ℓk + 1), we see that the function f (x) =

∞ X

(2ℓk + 1)tℓmk mk (x)

k=1

satisfies W f = 0 but f ∈ D ′ (G) is not smooth. Since the only non-zero triangular number which is also a cube is 1 (see, e.g., [17]) we see that the operator 2iD33 − L is globally hypoelliptic.

¨ HORMANDER CLASS Ψm (G) ON COMPACT LIE GROUPS

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2 2 Example 5.8. We consider P = D21 − D22 = − 21 (∂− + ∂+ ). The corresponding symbol satisfies p ℓ 2(σP )ℓmn = (ℓ + n)(ℓ + n − 1)(ℓ − n + 2)(ℓ − n + 1)δm,n−2 p ℓ + (ℓ + m)(ℓ + m − 1)(ℓ − m + 2)(ℓ − m + 1)δm−2,n .

Furthermore, by Leibniz’ rule we deduce D11 σP = D22 σP = 0 and D12 σP = −σ∂+ , D21 σP = −σ∂− . The symbol σP is invertible only for ℓ + 21 ∈ 2N0 . Therefore, P is neither elliptic nor does it satisfy subelliptic estimates. References [1] M. S. Agranovich, Elliptic pseudodifferential operators on a closed curve. (Russian) Trudy Moskov. Mat. Obshch., 47 (1984), 22–67. [2] R. Beals, Characterization of pseudo-differential operators and applications. Duke Math. J., 44 (1977), 45–57. [3] A. Bergamasco, S. Zani and S. Lu´ıs, Prescribing analytic singularities for solutions of a class of vector fields on the torus. Trans. Amer. Math. Soc., 357 (2005), 4159–4174. [4] A. Himonas and G. Petronilho, Global hypoellipticity and simultaneous approximability. J. Funct. Anal. 170 (2000), 356–365. [5] L. H¨ ormander, Pseudo-differential operators and hypoelliptic equations. Proc. Sympos. Pure Math., pp. 138–183. Amer. Math. Soc., 1967. [6] L. H¨ ormander, Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147–171. [7] M. Ruzhansky and V. Turunen, Pseudo-differential operators and symmetries, Birkh¨auser, Basel, 2010. [8] M. Ruzhansky and V. Turunen, Quantization of pseudo-differential operators on the torus, to appear in J. Fourier Anal. Appl., arXiv:0805.2892v1. [9] M. Ruzhansky and V. Turunen, Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere, preprint, arXiv:0812.3961v1. [10] S. A. Salamanca-Riba and D. A. Vogan Jr., On the classification of unitary representations of reductive Lie groups, Ann. of Math., 148 (1998), 1067–1133. [11] T. Sherman, Fourier analysis on the sphere, Trans. Amer. Math. Soc., 209 (1975), 1–31. [12] T. Sherman, The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one, Acta Math., 164 (1990), 73–144. [13] M. A. Shubin, Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin, 1987. [14] N. J. A. Sloane, Ed., The On-Line Encyclopedia of Integer Sequences, published electronically at www.research.att.com/~njas/sequences/, Sequence A001108. [15] M. Taylor, Noncommutative microlocal analysis. I, Mem. Amer. Math. Soc. 52 (1984), no. 313. [16] M. E. Taylor, Beals–Cordes -type characterizations of pseudodifferential operators. Proc. Amer. Math. Soc., 125 (1997), 1711–1716. [17] E. W. Weisstein, Cubic Triangular Number, from MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CubicTriangularNumber.html [18] H. Widom, A complete symbolic calculus for pseudodifferential operators. Bull. Sci. Math., 104 (1980), 19–63. Michael Ruzhansky: Department of Mathematics Imperial College London 180 Queen’s Gate, London SW7 2AZ United Kingdom E-mail address [email protected] Ville Turunen: Helsinki University of Technology

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MICHAEL RUZHANSKY, VILLE TURUNEN, AND JENS WIRTH

Institute of Mathematics P.O. Box 1100 FI-02015 HUT Finland E-mail address [email protected] Jens Wirth: Department of Mathematics Imperial College London 180 Queen’s Gate, London SW7 2AZ United Kingdom E-mail address [email protected]